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Chords of hemififths

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Below are listed the dyadic chords of 11-limit hemififths temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 441/440 werckismic, by 896/891 pentacircle, by 243/242 rastmic, and by 1344/1331 hemimin. Those requiring tempering by any two of 540/539, 441/440 or 243/242 are labeled jove, and those requiring both 441/440 and 896/891 are labeled pele. The label "nofives" refers to the unnamed rank-three temperament tempering out 243/242, 896/891 and 1344/1331, and if any two of these are needed the chord is so labled. "Nofives" refers to the fact that it is in essence a no-fives version of hemififths; if the full hemififths is required because of the tempering out of three independent hemififths commas, the chord is labeled "hemififths".

A striking feature of these hemififths chords is that essentially just chords tend to be of higher complexity than essentially tempered chords. Hemififths has MOS of size 7, 10, 17 and 24, and even seven notes are well-supplied with chords, mostly but by no means entirely essentially tempered chords. Extending consideration to the 13-limit adds even more such chords.

Triads

Number
 Chord
 Transversal
 Type
1
 0-1-2
 1-11/9-3/2
 rastmic
2
 0-1-3
 1-11/9-11/6
 utonal
3
 0-2-3
 1-3/2-11/6
 otonal
4
 0-1-4
 1-11/9-9/8
 rastmic
5
 0-2-4
 1-3/2-9/8
 ambitonal
6
 0-3-4
 1-11/6-9/8
 rastmic
7
 0-1-5
 1-11/9-11/8
 utonal
8
 0-2-5
 1-3/2-11/8
 otonal
9
 0-3-5
 1-11/6-11/8
 utonal
10
 0-4-5
 1-9/8-11/8
 otonal
11
 0-3-8
 1-11/6-14/11
 hemimin
12
 0-4-8
 1-9/8-14/11
 pentacircle
13
 0-5-8
 1-11/8-14/11
 hemimin
14
 0-1-9
 1-11/9-14/9
 otonal
15
 0-4-9
 1-9/8-14/9
 pentacircle
16
 0-5-9
 1-11/8-14/9
 pentacircle
17
 0-8-9
 1-14/11-14/9
 utonal
18
 0-2-11
 1-3/2-7/6
 otonal
19
 0-3-11
 1-11/6-7/6
 otonal
20
 0-8-11
 1-14/11-7/6
 utonal
21
 0-9-11
 1-14/9-7/6
 utonal
22
 0-1-12
 1-11/9-10/7
 swetismic
23
 0-3-12
 1-11/6-10/7
 swetismic
24
 0-4-12
 1-9/8-10/7
 werckismic
25
 0-8-12
 1-14/11-10/7
 werckismic
26
 0-9-12
 1-14/9-10/7
 swetismic
27
 0-11-12
 1-7/6-10/7
 swetismic
28
 0-1-13
 1-11/9-7/4
 werckismic
29
 0-2-13
 1-3/2-7/4
 otonal
30
 0-4-13
 1-9/8-7/4
 otonal
31
 0-5-13
 1-11/8-7/4
 otonal
32
 0-8-13
 1-14/11-7/4
 utonal
33
 0-9-13
 1-14/9-7/4
 utonal
34
 0-11-13
 1-7/6-7/4
 utonal
35
 0-12-13
 1-10/7-7/4
 werckismic
36
 0-8-20
 1-14/11-20/11
 otonal
37
 0-9-20
 1-14/9-20/11
 swetismic
38
 0-11-20
 1-7/6-20/11
 swetismic
39
 0-12-20
 1-10/7-20/11
 utonal
40
 0-1-21
 1-11/9-10/9
 otonal
41
 0-8-21
 1-14/11-10/9
 werckismic
42
 0-9-21
 1-14/9-10/9
 otonal
43
 0-12-21
 1-10/7-10/9
 utonal
44
 0-13-21
 1-7/4-10/9
 werckismic
45
 0-20-21
 1-20/11-10/9
 utonal
46
 0-2-23
 1-3/2-5/3
 otonal
47
 0-3-23
 1-11/6-5/3
 otonal
48
 0-11-23
 1-7/6-5/3
 otonal
49
 0-12-23
 1-10/7-5/3
 utonal
50
 0-20-23
 1-20/11-5/3
 utonal
51
 0-21-23
 1-10/9-5/3
 utonal
52
 0-2-25
 1-3/2-5/4
 otonal
53
 0-4-25
 1-9/8-5/4
 otonal
54
 0-5-25
 1-11/8-5/4
 otonal
55
 0-12-25
 1-10/7-5/4
 utonal
56
 0-13-25
 1-7/4-5/4
 otonal
57
 0-20-25
 1-20/11-5/4
 utonal
58
 0-21-25
 1-10/9-5/4
 utonal
59
 0-23-25
 1-5/3-5/4
 utonal

Tetrads

Number
 Chord
 Transversal
 Type
1
 0-1-2-3
 1-11/9-3/2-11/6
 rastmic
2
 0-1-2-4
 1-11/9-3/2-9/8
 rastmic
3
 0-1-3-4
 1-11/9-11/6-9/8
 rastmic
4
 0-2-3-4
 1-3/2-11/6-9/8
 rastmic
5
 0-1-2-5
 1-11/9-3/2-11/8
 rastmic
6
 0-1-3-5
 1-11/9-11/6-11/8
 utonal
7
 0-2-3-5
 1-3/2-11/6-11/8
 ambitonal
8
 0-1-4-5
 1-11/9-9/8-11/8
 rastmic
9
 0-2-4-5
 1-3/2-9/8-11/8
 otonal
10
 0-3-4-5
 1-11/6-9/8-11/8
 rastmic
11
 0-3-4-8
 1-11/6-9/8-14/11
 nofives
12
 0-3-5-8
 1-11/6-11/8-14/11
 hemimin
13
 0-4-5-8
 1-9/8-11/8-14/11
 nofives
14
 0-1-4-9
 1-11/9-9/8-14/9
 nofives
15
 0-1-5-9
 1-11/9-11/8-14/9
 pentacircle
16
 0-4-5-9
 1-9/8-11/8-14/9
 pentacircle
17
 0-4-8-9
 1-9/8-14/11-14/9
 pentacircle
18
 0-5-8-9
 1-11/8-14/11-14/9
 nofives
19
 0-2-3-11
 1-3/2-11/6-7/6
 otonal
20
 0-3-8-11
 1-11/6-14/11-7/6
 hemimin
21
 0-8-9-11
 1-14/11-14/9-7/6
 utonal
22
 0-1-3-12
 1-11/9-11/6-10/7
 swetismic
23
 0-1-4-12
 1-11/9-9/8-10/7
 jove
24
 0-3-4-12
 1-11/6-9/8-10/7
 jove
25
 0-3-8-12
 1-11/6-14/11-10/7
 hemififths
26
 0-4-8-12
 1-9/8-14/11-10/7
 pele
27
 0-1-9-12
 1-11/9-14/9-10/7
 swetismic
28
 0-4-9-12
 1-9/8-14/9-10/7
 hemififths
29
 0-8-9-12
 1-14/11-14/9-10/7
 jove
30
 0-3-11-12
 1-11/6-7/6-10/7
 swetismic
31
 0-8-11-12
 1-14/11-7/6-10/7
 jove
32
 0-9-11-12
 1-14/9-7/6-10/7
 swetismic
33
 0-1-2-13
 1-11/9-3/2-7/4
 jove
34
 0-1-4-13
 1-11/9-9/8-7/4
 jove
35
 0-2-4-13
 1-3/2-9/8-7/4
 otonal
36
 0-1-5-13
 1-11/9-11/8-7/4
 werckismic
37
 0-2-5-13
 1-3/2-11/8-7/4
 otonal
38
 0-4-5-13
 1-9/8-11/8-7/4
 otonal
39
 0-4-8-13
 1-9/8-14/11-7/4
 pentacircle
40
 0-5-8-13
 1-11/8-14/11-7/4
 hemimin
41
 0-1-9-13
 1-11/9-14/9-7/4
 werckismic
42
 0-4-9-13
 1-9/8-14/9-7/4
 pentacircle
43
 0-5-9-13
 1-11/8-14/9-7/4
 pentacircle
44
 0-8-9-13
 1-14/11-14/9-7/4
 utonal
45
 0-2-11-13
 1-3/2-7/6-7/4
 ambitonal
46
 0-8-11-13
 1-14/11-7/6-7/4
 utonal
47
 0-9-11-13
 1-14/9-7/6-7/4
 utonal
48
 0-1-12-13
 1-11/9-10/7-7/4
 jove
49
 0-4-12-13
 1-9/8-10/7-7/4
 werckismic
50
 0-8-12-13
 1-14/11-10/7-7/4
 werckismic
51
 0-9-12-13
 1-14/9-10/7-7/4
 jove
52
 0-11-12-13
 1-7/6-10/7-7/4
 jove
53
 0-8-9-20
 1-14/11-14/9-20/11
 swetismic
54
 0-8-11-20
 1-14/11-7/6-20/11
 swetismic
55
 0-9-11-20
 1-14/9-7/6-20/11
 swetismic
56
 0-8-12-20
 1-14/11-10/7-20/11
 werckismic
57
 0-9-12-20
 1-14/9-10/7-20/11
 swetismic
58
 0-11-12-20
 1-7/6-10/7-20/11
 swetismic
59
 0-1-9-21
 1-11/9-14/9-10/9
 otonal
60
 0-8-9-21
 1-14/11-14/9-10/9
 werckismic
61
 0-1-12-21
 1-11/9-10/7-10/9
 swetismic
62
 0-8-12-21
 1-14/11-10/7-10/9
 werckismic
63
 0-9-12-21
 1-14/9-10/7-10/9
 swetismic
64
 0-1-13-21
 1-11/9-7/4-10/9
 werckismic
65
 0-8-13-21
 1-14/11-7/4-10/9
 werckismic
66
 0-9-13-21
 1-14/9-7/4-10/9
 werckismic
67
 0-12-13-21
 1-10/7-7/4-10/9
 werckismic
68
 0-8-20-21
 1-14/11-20/11-10/9
 werckismic
69
 0-9-20-21
 1-14/9-20/11-10/9
 swetismic
70
 0-12-20-21
 1-10/7-20/11-10/9
 utonal
71
 0-2-3-23
 1-3/2-11/6-5/3
 otonal
72
 0-2-11-23
 1-3/2-7/6-5/3
 otonal
73
 0-3-11-23
 1-11/6-7/6-5/3
 otonal
74
 0-3-12-23
 1-11/6-10/7-5/3
 swetismic
75
 0-11-12-23
 1-7/6-10/7-5/3
 swetismic
76
 0-11-20-23
 1-7/6-20/11-5/3
 swetismic
77
 0-12-20-23
 1-10/7-20/11-5/3
 utonal
78
 0-12-21-23
 1-10/7-10/9-5/3
 utonal
79
 0-20-21-23
 1-20/11-10/9-5/3
 utonal
80
 0-2-4-25
 1-3/2-9/8-5/4
 otonal
81
 0-2-5-25
 1-3/2-11/8-5/4
 otonal
82
 0-4-5-25
 1-9/8-11/8-5/4
 otonal
83
 0-4-12-25
 1-9/8-10/7-5/4
 werckismic
84
 0-2-13-25
 1-3/2-7/4-5/4
 otonal
85
 0-4-13-25
 1-9/8-7/4-5/4
 otonal
86
 0-5-13-25
 1-11/8-7/4-5/4
 otonal
87
 0-12-13-25
 1-10/7-7/4-5/4
 werckismic
88
 0-12-20-25
 1-10/7-20/11-5/4
 utonal
89
 0-12-21-25
 1-10/7-10/9-5/4
 utonal
90
 0-13-21-25
 1-7/4-10/9-5/4
 werckismic
91
 0-20-21-25
 1-20/11-10/9-5/4
 utonal
92
 0-2-23-25
 1-3/2-5/3-5/4
 ambitonal
93
 0-12-23-25
 1-10/7-5/3-5/4
 utonal
94
 0-20-23-25
 1-20/11-5/3-5/4
 utonal
95
 0-21-23-25
 1-10/9-5/3-5/4
 utonal

Pentads

Number
 Chord
 Transversal
 Type
1
 0-1-2-3-4
 1-11/9-3/2-11/6-9/8
 rastmic
2
 0-1-2-3-5
 1-11/9-3/2-11/6-11/8
 rastmic
3
 0-1-2-4-5
 1-11/9-3/2-9/8-11/8
 rastmic
4
 0-1-3-4-5
 1-11/9-11/6-9/8-11/8
 rastmic
5
 0-2-3-4-5
 1-3/2-11/6-9/8-11/8
 rastmic
6
 0-3-4-5-8
 1-11/6-9/8-11/8-14/11
 nofives
7
 0-1-4-5-9
 1-11/9-9/8-11/8-14/9
 nofives
8
 0-4-5-8-9
 1-9/8-11/8-14/11-14/9
 nofives
9
 0-1-3-4-12
 1-11/9-11/6-9/8-10/7
 jove
10
 0-3-4-8-12
 1-11/6-9/8-14/11-10/7
 hemififths
11
 0-1-4-9-12
 1-11/9-9/8-14/9-10/7
 hemififths
12
 0-4-8-9-12
 1-9/8-14/11-14/9-10/7
 hemififths
13
 0-3-8-11-12
 1-11/6-14/11-7/6-10/7
 hemififths
14
 0-8-9-11-12
 1-14/11-14/9-7/6-10/7
 jove
15
 0-1-2-4-13
 1-11/9-3/2-9/8-7/4
 jove
16
 0-1-2-5-13
 1-11/9-3/2-11/8-7/4
 jove
17
 0-1-4-5-13
 1-11/9-9/8-11/8-7/4
 jove
18
 0-2-4-5-13
 1-3/2-9/8-11/8-7/4
 otonal
19
 0-4-5-8-13
 1-9/8-11/8-14/11-7/4
 nofives
20
 0-1-4-9-13
 1-11/9-9/8-14/9-7/4
 hemififths
21
 0-1-5-9-13
 1-11/9-11/8-14/9-7/4
 pele
22
 0-4-5-9-13
 1-9/8-11/8-14/9-7/4
 pentacircle
23
 0-4-8-9-13
 1-9/8-14/11-14/9-7/4
 pentacircle
24
 0-5-8-9-13
 1-11/8-14/11-14/9-7/4
 nofives
25
 0-8-9-11-13
 1-14/11-14/9-7/6-7/4
 utonal
26
 0-1-4-12-13
 1-11/9-9/8-10/7-7/4
 jove
27
 0-4-8-12-13
 1-9/8-14/11-10/7-7/4
 pele
28
 0-1-9-12-13
 1-11/9-14/9-10/7-7/4
 jove
29
 0-4-9-12-13
 1-9/8-14/9-10/7-7/4
 hemififths
30
 0-8-9-12-13
 1-14/11-14/9-10/7-7/4
 jove
31
 0-8-11-12-13
 1-14/11-7/6-10/7-7/4
 jove
32
 0-9-11-12-13
 1-14/9-7/6-10/7-7/4
 jove
33
 0-8-9-11-20
 1-14/11-14/9-7/6-20/11
 swetismic
34
 0-8-9-12-20
 1-14/11-14/9-10/7-20/11
 jove
35
 0-8-11-12-20
 1-14/11-7/6-10/7-20/11
 jove
36
 0-9-11-12-20
 1-14/9-7/6-10/7-20/11
 swetismic
37
 0-1-9-12-21
 1-11/9-14/9-10/7-10/9
 swetismic
38
 0-8-9-12-21
 1-14/11-14/9-10/7-10/9
 jove
39
 0-1-9-13-21
 1-11/9-14/9-7/4-10/9
 werckismic
40
 0-8-9-13-21
 1-14/11-14/9-7/4-10/9
 werckismic
41
 0-1-12-13-21
 1-11/9-10/7-7/4-10/9
 jove
42
 0-8-12-13-21
 1-14/11-10/7-7/4-10/9
 werckismic
43
 0-9-12-13-21
 1-14/9-10/7-7/4-10/9
 jove
44
 0-8-9-20-21
 1-14/11-14/9-20/11-10/9
 jove
45
 0-8-12-20-21
 1-14/11-10/7-20/11-10/9
 werckismic
46
 0-9-12-20-21
 1-14/9-10/7-20/11-10/9
 swetismic
47
 0-2-3-11-23
 1-3/2-11/6-7/6-5/3
 otonal
48
 0-3-11-12-23
 1-11/6-7/6-10/7-5/3
 swetismic
49
 0-11-12-20-23
 1-7/6-10/7-20/11-5/3
 swetismic
50
 0-12-20-21-23
 1-10/7-20/11-10/9-5/3
 utonal
51
 0-2-4-5-25
 1-3/2-9/8-11/8-5/4
 otonal
52
 0-2-4-13-25
 1-3/2-9/8-7/4-5/4
 otonal
53
 0-2-5-13-25
 1-3/2-11/8-7/4-5/4
 otonal
54
 0-4-5-13-25
 1-9/8-11/8-7/4-5/4
 otonal
55
 0-4-12-13-25
 1-9/8-10/7-7/4-5/4
 werckismic
56
 0-12-13-21-25
 1-10/7-7/4-10/9-5/4
 werckismic
57
 0-12-20-21-25
 1-10/7-20/11-10/9-5/4
 utonal
58
 0-12-20-23-25
 1-10/7-20/11-5/3-5/4
 utonal
59
 0-12-21-23-25
 1-10/7-10/9-5/3-5/4
 utonal
60
 0-20-21-23-25
 1-20/11-10/9-5/3-5/4
 utonal

Hexads

Number
 Chord
 Transversal
 Type
1
 0-1-2-3-4-5
 1-11/9-3/2-11/6-9/8-11/8
 rastmic
2
 0-1-2-4-5-13
 1-11/9-3/2-9/8-11/8-7/4
 jove
3
 0-1-4-5-9-13
 1-11/9-9/8-11/8-14/9-7/4
 hemififths
4
 0-4-5-8-9-13
 1-9/8-11/8-14/11-14/9-7/4
 nofives
5
 0-1-4-9-12-13
 1-11/9-9/8-14/9-10/7-7/4
 hemififths
6
 0-4-8-9-12-13
 1-9/8-14/11-14/9-10/7-7/4
 hemififths
7
 0-8-9-11-12-13
 1-14/11-14/9-7/6-10/7-7/4
 jove
8
 0-8-9-11-12-20
 1-14/11-14/9-7/6-10/7-20/11
 jove
9
 0-1-9-12-13-21
 1-11/9-14/9-10/7-7/4-10/9
 jove
10
 0-8-9-12-13-21
 1-14/11-14/9-10/7-7/4-10/9
 jove
11
 0-8-9-12-20-21
 1-14/11-14/9-10/7-20/11-10/9
 jove
12
 0-2-4-5-13-25
 1-3/2-9/8-11/8-7/4-5/4
 otonal
13
 0-12-20-21-23-25
 1-10/7-20/11-10/9-5/3-5/4
 utonal